Nonuniqueness in nonlinear heat propagation: a heat wave coming from infinity

Resumen

. It is well known that nonnegative solutions of the heat equation defined in a stripQ = RN ⇥ (0, T) are uniquely determined by their initial data. In this paper we constructa nonnegative solution H(x,t) of the nonlinear heat equationut = (um)xx up, 1 < p < m, (E)which takes on zero initial data, H(x, 0) ⌘ 0, and is nontrivial for t ⌧ > 0. This specialsolution has the following properties: (i) H(x,t) is bounded for t ⌧ > 0. (ii) There existsa constant r > 0 such thatH(x,t) > 0 if rt < x < 1,H(x,t) = 0 if 1 < x < rt,where = (m p)/2(p 1). (iii) It has the self-similar form H(x,t) = t↵f(xt), where0 f(⌘) c is supported in the interval r ⌘ < 1. A similar phenomenon holds in severalspace dimensions with radial symmetry, where it describes a form of focussing. On the otherhand, such a nonuniqueness does not occur for other ranges of m and p. In particular, itdoes not happen for linear di↵usion, m = 1.